One thing that can be helpful for sleepy babies is for them to spend lots
of time skin to skin with mom. I often suggest that baby be unwrapped and
dressed in a diaper, and mom can wear baby next to her skin. She can do
this in bed and cover them both with the bed covers, or she can use a
front opening shirt and cover up with a baby blanket, shawl, or use a
front pack that is open between the baby's skin and mom under a large
shirt perhaps. I find that skin to skin contact can be great for
stimulating suckling.

In the case of this infant, I think cup feeding the mothers expressed
milk, or using a feeding tube device would be in order as it sounds like
this baby needs to be taking in more. Also, why this baby is so sleepy
needs to be explored. A good phyical exam is in order and a suck
evaluation. Did the mom have a long labor, traumatic birth, or
medications in labor?  Is the mom keeping the baby wrapped very tightly
and warmly? I have noticed that babies kept overly warm can be very sleepy.

I also find it helpful to teach moms about baby's sleep cycles and cues
so they can learn to distinquish when baby is in a deep sleep vs a light
sleep and more easily roused.

Finally, for the baby who falls asleep quickly after latching on, it may
be helpful to teach mom to take baby off once he is not actively nursing,
rouse the baby and then put baby back to breast and work on keeping baby
awake longer to nurse. It is possible that once this baby gets some extra
mom's milk on board, that he may feed more effectively unless there is
some other problem that is also interfering.

Ione Sims, GNM, IBCLC